Let M be a n-dimensional complex manifold, let S be a globally irreducible compact analytic hypersurface with regular part S'=S-Sing(S), and let (f,g) be a pair of distinct holomorphic self-maps coinciding on S and such that g is a local biholomorphism over an open neighborhood of S'. We show that under certain hypotheses, on the pair (f,g) or on the way S' sits into M, we are able to define a 1-dimensional holomorphic foliation on S' and related partial holomorphic connections on some holomorphic vector bundles over S'. Consequently, we can obtain index theorems using the so-called Lehmann-Suwa machinery, which is based on localization of characteristic classes in Cech-de Rham cohomology.